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Course material availale online at www2.ph.ed.ac.uk/~dmarendu/ASP.html

4 tutorial sheets to be spread between 6-8 tutorial sessions, beginning in around 2 weeks. Tutorials to be held on Wednesday, 2-4 pm, in room 6309.

Course lectured by Davide Marenduzzo (2506 JCMB).

\section{Course Outline}
Recap of basic statistical mechanics, includng the Ising model.
Mean field theory of phase transitions.
Renormalisation group, used to treat phase transitions more accurately.
Non-equilibrium phenomena (approach to liquid helium).

\emph{Phase transitions} refer to \emph{abrupt} changes in ordering or behaviour of a system. Some examples of phase transitions are:

\emph{Order-disorder} transitions are a very important type of transition characterised by an \emph{order parameter}, for example the total magnetisation of an Ising lattice, which breaks away from zero in an equilibrium phase transition.

The Vicsek model has been used to model the dynamics of bird flocking, and exhibits a non-equilibrium transition between ordered (clustered flocks) and disordered (independent `movers') phases.

Both Ising and Vicsek models exhibit a key feature of phase transitions; near the critical point the system undergoes massive fluctuations between ordered and disordered phases.

\emph{Localisation/delocalisation} transitions are observe to occur when DNA molecules are exposed to elevated temperatures, and become denatured. At low temperatures DNA strand are `zipped' together in pairs, and around a certain critical temperature they undergo a second-order phase transition and unzip, drifting apart.

\section{Statistical Mechanics Recap}

Let $\H$ be the Hamiltonian, or total energy, of a system.

The partition function is the sum over all Boltzmann probabilities $$Z_n \equiv \sum_\mathrm{configurations} e^{-\beta \H(\mathrm{configuration})}$$ where $\beta \equiv \frac{1}{k_B T}$. The partition function encodes all of the thermodynamic information about the system. The expectation value of a thermodynamic quantity $Q$ is calculated according to $$\langle Q \rangle = \frac{ \sum_\mathrm{configurations} Q(\mathrm{conf}) e^{-\beta \H(\mathrm{conf})} }{ \sum_\mathrm{configurations} e^{-\beta \mathcal{H}(\mathrm{conf})} }$$. We identify the sum in the denominator with the partition function $Z_N$, and use the following trick to 

The Helmholtz free energy $F = - k_B T \log Z_N$ and the following thermodynamic quantities follow:

$p = - \pdiff{F}{V}$

$\pdiff{F}{T} = -S$

$\pdiffsq{}{\beta} \log Z_N$

$\pdiffsq{\log Z_N}{\beta} = \frac{1}{Z_N} \pdiffsq{Z_N}{\beta} - \left( \frac{ \pdiff{Z_N}{\beta}}{Z_N} \right) = \frac{ \sum_\mathrm{conf} \mathcal{H}^2 e^{-\beta \mathcal{H}} }{ \sum_\mathrm{conf} e^{-\beta \mathcal{H}} } - \langle \mathcal{H} \rangle^2$

$k_B T^2 C_V = \langle \mathcal{H}^2 \rangle - \langle \mathcal{H} \rangle^2$

This is a fluctuation-dissipation relation.

\section{Uncorrelated Systems}

Consider a lattice of spins $\sigma_i = \pm 1$, $i \in [1,N]$.

$- \beta \H = \sum_{i=1}^N \beta h$ where $h$ is an external magnetic field, is called the \emph{action}, and describes a completely uncorrelated system (i.e. there is no iteraction between the spins. The partition function is

$$Z_N = \sum_{\sigma_1 = \pm 1} ... \sum_{\sigma_N = \pm 1} e^{+\beta \sum h \sigma_i} = \left( \sum_{\sigma_1 = \pm 1} e^{\H \sigma_1} \right) ... \left( \sum_{\sigma_N = \pm 1} e^{\H \sigma_N} \right) = \left( \sum_{\sigma = \pm 1} e^{\H \sigma} \right)^N = (e^H + e^{-\H})^N = (2 \cosh \H)^N$$

which factorises, since the spins are uncorrelated.

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